The paper is devoted to the nonlinear hyperbolic Cauchy problem \(\partial_{tt}u(t,x)-(1+\int_{R^n}dy K(x-y)|\nabla_yu(t,y)|^2)\Delta_xu(t,x)=0\), \(u(0,x)=\varepsilon u_0(x)\), \((\partial_tu)(0,x)=\varepsilon u_1(x)\), \(\varepsilon>0\), \(t\geq 0\), \(x\in\mathbb{R}^n\), \((u_0,u_1)\in C_c^\infty(\mathbb{R}^n\times\mathbb{R}^n)\), \(n>3\). The kernel \(z\to K(z)\) is a positive smooth function rapidly decreasing as \(|z|\to \infty\). The author proves a global existence and uniqueness result using the generalized energy estimates combined with von Wahl inequalities. The contraction method and the continuation principle for differential equations are applied to complete the proof of the existence and uniqueness theorem.